We propose generalized portmanteau-type test statistics in the frequency domain to test independence between two stationary time series. The test statistics are formed analogous to the one in Chen and Deo (2004, Econometric Theory 20, 382-416), who extended the applicability of portmanteau goodness-of-fit test to the long memory case. Under the null hypothesis of independence, the asymptotic standard normal distributions of the proposed statistics are derived under fairly mild conditions. In particular, each time series is allowed to possess short memory, long memory or anti-persistence. A simulation study shows that the tests have reasonable size and power properties.
The Portmanteau test provides the vanilla method for detecting serial correlations in classical univariate time series analysis. The method is extended to the case of observations from a locally stationary functional time series. Asymptotic critical values are obtained by a suitable block multiplier bootstrap procedure. The test is shown to asymptotically hold its level and to be consistent against general alternatives.
A new portmanteau test statistic is proposed for detecting nonlinearity in time series data. In this paper, we elaborate on the Toeplitz autocorrelation matrix to the autocorrelation and cross-correlation of residuals and squared residuals block matrix. We derive a new portmanteau test statistic using the log of the determinant of the mth autocorrelations and cross-correlations block matrix. The asymptotic distribution of the proposed test statistic is derived as a linear combination of chi-squared distributions and can be approximated by a gamma distribution. This test is applied to identify …
We consider a zero mean discrete time series, and define its discrete Fourier transform at the canonical frequencies. It is well known that the discrete Fourier transform is asymptotically uncorrelated at the canonical frequencies if and if only the time series is second order stationary. Exploiting this important property, we construct a Portmanteau type test statistic for testing stationarity of the time series. It is shown that under the null of stationarity, the test statistic is approximately a chi square distribution. To examine the power of the test statistic, the asymptotic distributio…
We propose a specification test for conditional location--scale models based on extremal dependence properties of the standardized residuals. We do so comparing the left-over serial extremal dependence -- as measured by the pre-asymptotic tail copula -- with that arising under serial independence at different lags. Our main theoretical results show that the proposed Portmanteau-type test statistics have nuisance parameter-free asymptotic limits. The test statistics are easy to compute, as they only depend on the standardized residuals, and critical values are likewise easily obtained from the …
This paper proposes some novel one-sided omnibus tests for independence between two multivariate stationary time series. These new tests apply the Hilbert-Schmidt independence criterion (HSIC) to test the independence between the innovations of both time series. Under regular conditions, the limiting null distributions of our HSIC-based tests are established. Next, our HSIC-based tests are shown to be consistent. Moreover, a residual bootstrap method is used to obtain the critical values for our HSIC-based tests, and its validity is justified. Compared with the existing cross-correlation-based…
Two-sample and independence tests with the kernel-based MMD and HSIC have shown remarkable results on i.i.d. data and stationary random processes. However, these statistics are not directly applicable to non-stationary random processes, a prevalent form of data in many scientific disciplines. In this work, we extend the application of MMD and HSIC to non-stationary settings by assuming access to independent realisations of the underlying random process. These realisations - in the form of non-stationary time-series measured on the same temporal grid - can then be viewed as i.i.d. samples from …
We consider strictly stationary heavy tailed time series whose finite-dimensional exponent measures are concentrated on axes, and hence their extremal properties cannot be tackled using classical multivariate regular variation that is suitable for time series with extremal dependence. We recover relevant information about limiting behavior of time series with extremal independence by introducing a sequence of scaling functions and conditional scaling exponent. Both quantities provide more information about joint extremes than a widely used tail dependence coefficient. We calculate the scaling …
The volatility modeling for autoregressive univariate time series is considered. A benchmark approach is the stationary ARCH model of Engle (1982). Motivated by real data evidence, processes with non constant unconditional variance and ARCH effects have been recently introduced. We take into account such possible non stationarity and propose simple testing procedures for ARCH effects. Adaptive McLeod and Li's portmanteau and ARCH-LM tests for checking for second order dynamics are provided. The standard versions of these tests, commonly used by practitioners, suppose constant unconditional var…
We propose a method for inferring the conditional independence graph (CIG) of a high-dimensional Gaussian vector time series (discrete-time process) from a finite-length observation. By contrast to existing approaches, we do not rely on a parametric process model (such as, e.g., an autoregressive model) for the observed random process. Instead, we only require certain smoothness properties (in the Fourier domain) of the process. The proposed inference scheme works even for sample sizes much smaller than the number of scalar process components if the true underlying CIG is sufficiently sparse. …
Given a heterogeneous time-series sample, the objective is to find points in time (called change points) where the probability distribution generating the data has changed. The data are assumed to have been generated by arbitrary unknown stationary ergodic distributions. No modelling, independence or mixing assumptions are made. A novel, computationally efficient, nonparametric method is proposed, and is shown to be asymptotically consistent in this general framework. The theoretical results are complemented with experimental evaluations.
We describe a family of conservative statistical tests for independence of two autocorrelated time series. The series may take values in any sets, and one of them must be stationary. A user-specified function quantifying the association of a segment of the two series is compared to an ensemble obtained by time-shifting the stationary series -N to N steps. If the series are independent, the unshifted value is in the top m shifted values with probability at most m/(N+1). For large N, the probability approaches m/(2N+1). A conservative test rejects independence at significance α if the unshifted …
We formulate nonparametric and semiparametric hypothesis testing of multivariate stationary linear time series in a unified fashion and propose new test statistics based on estimators of the spectral density matrix. The limiting distributions of these test statistics under null hypotheses are always normal distributions, and they can be implemented easily for practical use. If null hypotheses are false, as the sample size goes to infinity, they diverge to infinity and consequently are consistent tests for any alternative. The approach can be applied to various null hypotheses such as the indep…
The covariance function and the variogram play very important roles in modelling and in prediction of spatial and spatio-temporal data. The assumption of second order stationarity, in space and time, is often made in the analysis of spatial data and the spatio-temporal data. Several times the assumption of stationarity is considered to be very restrictive, and therefore, a weaker assumption that the data is Intrinsically stationary both in space and time is often made and used, mainly by the geo-statisticians and other environmental scientists. In this paper we consider the data to be intrinsi…
Stationarity is a very general, qualitative assumption, that can be assessed on the basis of application specifics. It is thus a rather attractive assumption to base statistical analysis on, especially for problems for which less general qualitative assumptions, such as independence or finite memory, clearly fail. However, it has long been considered too general to allow for statistical inference to be made. One of the reasons for this is that rates of convergence, even of frequencies to the mean, are not available under this assumption alone. Recently, it has been shown that, while some natur…
We consider the problem of finding confidence intervals for the risk of forecasting the future of a stationary, ergodic stochastic process, using a model estimated from the past of the process. We show that a bootstrap procedure provides valid confidence intervals for the risk, when the data source is sufficiently mixing, and the loss function and the estimator are suitably smooth. Autoregressive (AR(d)) models estimated by least squares obey the necessary regularity conditions, even when mis-specified, and simulations show that the finite- sample coverage of our bounds quickly converges to th…
The independence clustering problem is considered in the following formulation: given a set $S$ of random variables, it is required to find the finest partitioning $\{U_1,\dots,U_k\}$ of $S$ into clusters such that the clusters $U_1,\dots,U_k$ are mutually independent. Since mutual independence is the target, pairwise similarity measurements are of no use, and thus traditional clustering algorithms are inapplicable. The distribution of the random variables in $S$ is, in general, unknown, but a sample is available. Thus, the problem is cast in terms of time series. Two forms of sampling are con…
We consider the problem of inferring the conditional independence graph (CIG) of a multivariate stationary dicrete-time Gaussian random process based on a finite length observation. Using information-theoretic methods, we derive a lower bound on the error probability of any learning scheme for the underlying process CIG. This bound, in turn, yields a minimum required sample-size which is necessary for any algorithm regardless of its computational complexity, to reliably select the true underlying CIG. Furthermore, by analysis of a simple selection scheme, we show that the information-theoretic…
The family of U-statistics plays a fundamental role in statistics. This paper proves a novel exponential inequality for U-statistics under the time series setting. Explicit mixing conditions are given for guaranteeing fast convergence, the bound proves to be analogous to the one under independence, and extension to non-stationary time series is straightforward. The proof relies on a novel decomposition of U-statistics via exploiting the temporal correlatedness structure. Such results are of interest in many fields where high dimensional time series data are present. In particular, applications…
We consider a stationary spatio-temporal random process and assume that we have a sample. By defining a sequence of discrete Fourier transforms at canonical frequencies at each location, and using these complex valued random varables as observed sample, we obtain expressions for the spatio-temporal covariance functions and the spectral density functions of the spatio-temporal random processes. These spectra correspond to non separable class of random processes. The spatio-temporal covariance functions, obtained here are functions of the spatial distance and the temporal frequency and are simil…
In this paper we consider portmanteau tests for testing the adequacy of multiplicative seasonal autoregressive moving-average (SARMA) models under the assumption that the errors are uncorrelated but not necessarily independent.We relax the standard independence assumption on the error term in order to extend the range of application of the SARMA models.We study the asymptotic distributions of residual and normalized residual empirical autocovariances and autocorrelations underweak assumptions on the noise. We establish the asymptotic behaviour of the proposed statistics. A set of Monte Carlo e…
The modeling of spatio-temporal trends in temperature extremes can help better understand the structure and frequency of heatwaves in a changing climate. Here, we study annual temperature maxima over Southern Europe using a century-spanning dataset observed at 44 monitoring stations. Extending the spectral representation of max-stable processes, our modeling framework relies on a novel construction of max-infinitely divisible processes, which include covariates to capture spatio-temporal non-stationarities. Our new model keeps a popular max-stable process on the boundary of the parameter space…
Carnegie Mellon University（卡内基梅隆大学）；Santa Fe Institute（圣塔菲研究所）
The literature on statistical learning for time series assumes the asymptotic independence or ``mixing' of the data-generating process. These mixing assumptions are never tested, nor are there methods for estimating mixing rates from data. We give an estimator for the $β$-mixing rate based on a single stationary sample path and show it is $L_1$-risk consistent.
We consider stationary time series $\{X_j, j \in Z\} whose finite dimensional distributions are regularly varying with extremal independence. We assume that for each $h \geq 1$, conditionally on $X_0$ to exceed a threshold tending to infinity, the conditional distribution of $X_h$ suitably normalized converges weakly to a non degenerate distribution. We consider in this paper the estimation of the normalization and of the limiting distribution.
Measures of linear dependence (coherence) and nonlinear dependence (phase synchronization) between any number of multivariate time series are defined. The measures are expressed as the sum of lagged dependence and instantaneous dependence. The measures are non-negative, and take the value zero only when there is independence of the pertinent type. These measures are defined in the frequency domain and are applicable to stationary and non-stationary time series. These new results extend and refine significantly those presented in a previous technical report (Pascual-Marqui 2007, arXiv:0706.1776…
Graphical interaction models have become an important tool for analysing multivariate time series. In these models, the interrelationships among the components of a time series are described by undirected graphs in which the vertices depict the components while the edges indictate possible dependencies between the components. Current methods for the identification of the graphical structure are based on nonparametric spectral stimation, which prevents application of common model selection strategies. In this paper, we present a parametric approach for graphical interaction modelling of multiva…
The Rényi entropy is a generalization of the Shannon entropy and is widely used in mathematical statistics and applied sciences for quantifying the uncertainty in a probability distribution. We consider estimation of the quadratic Rényi entropy and related functionals for the marginal distribution of a stationary m-dependent sequence. The U-statistic estimators under study are based on the number of epsilon-close vector observations in the corresponding sample. A variety of asymptotic properties for these estimators are obtained (e.g., consistency, asymptotic normality, Poisson convergence). T…
We consider kernel estimation of marginal densities and regression functions of stationary processes. It is shown that for a wide class of time series, with proper centering and scaling, the maximum deviations of kernel density and regression estimates are asymptotically Gumbel. Our results substantially generalize earlier ones which were obtained under independence or beta mixing assumptions. The asymptotic results can be applied to assess patterns of marginal densities or regression functions via the construction of simultaneous confidence bands for which one can perform goodness-of-fit test…
Given a discrete-valued sample $X_1,...,X_n$ we wish to decide whether it was generated by a distribution belonging to a family $H_0$, or it was generated by a distribution belonging to a family $H_1$. In this work we assume that all distributions are stationary ergodic, and do not make any further assumptions (e.g. no independence or mixing rate assumptions). We would like to have a test whose probability of error (both Type I and Type II) is uniformly bounded. More precisely, we require that for each $ε$ there exist a sample size $n$ such that probability of error is upper-bounded by $ε$ for…
